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Properties

Label 1548.2.f.b.773.3
Level $1548$
Weight $2$
Character 1548.773
Analytic conductor $12.361$
Analytic rank $0$
Dimension $8$
CM discriminant -43
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1548,2,Mod(773,1548)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1548.773"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1548, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1548 = 2^{2} \cdot 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1548.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.3608422329\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3}, \sqrt{43})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 38x^{6} - 12x^{5} + 655x^{4} - 288x^{3} - 5034x^{2} + 4428x + 21222 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{43}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 773.3
Root \(2.41269 - 1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 1548.773
Dual form 1548.2.f.b.773.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.66312i q^{11} -5.87680 q^{13} +5.03990i q^{17} +8.14573i q^{23} -5.00000 q^{25} +4.51551 q^{31} +8.92799i q^{41} -6.55744 q^{43} +12.1020i q^{47} +7.00000 q^{49} -14.2542i q^{53} +3.61676i q^{59} -16.2691 q^{67} -13.1149 q^{79} +1.93408i q^{83} -16.5367 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 40 q^{25} + 56 q^{49} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1548\mathbb{Z}\right)^\times\).

\(n\) \(173\) \(433\) \(775\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.66312i − 0.802960i −0.915868 − 0.401480i \(-0.868496\pi\)
0.915868 − 0.401480i \(-0.131504\pi\)
\(12\) 0 0
\(13\) −5.87680 −1.62993 −0.814965 − 0.579510i \(-0.803244\pi\)
−0.814965 + 0.579510i \(0.803244\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.03990i 1.22236i 0.791493 + 0.611178i \(0.209304\pi\)
−0.791493 + 0.611178i \(0.790696\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.14573i 1.69850i 0.527989 + 0.849251i \(0.322946\pi\)
−0.527989 + 0.849251i \(0.677054\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 4.51551 0.811009 0.405505 − 0.914093i \(-0.367096\pi\)
0.405505 + 0.914093i \(0.367096\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.92799i 1.39432i 0.716916 + 0.697159i \(0.245553\pi\)
−0.716916 + 0.697159i \(0.754447\pi\)
\(42\) 0 0
\(43\) −6.55744 −1.00000
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.1020i 1.76526i 0.470064 + 0.882632i \(0.344231\pi\)
−0.470064 + 0.882632i \(0.655769\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 14.2542i − 1.95797i −0.203936 − 0.978984i \(-0.565373\pi\)
0.203936 − 0.978984i \(-0.434627\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.61676i 0.470863i 0.971891 + 0.235431i \(0.0756503\pi\)
−0.971891 + 0.235431i \(0.924350\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −16.2691 −1.98759 −0.993793 − 0.111241i \(-0.964517\pi\)
−0.993793 + 0.111241i \(0.964517\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.1149 −1.47554 −0.737769 − 0.675053i \(-0.764121\pi\)
−0.737769 + 0.675053i \(0.764121\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.93408i 0.212292i 0.994351 + 0.106146i \(0.0338511\pi\)
−0.994351 + 0.106146i \(0.966149\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.5367 −1.67905 −0.839525 − 0.543321i \(-0.817167\pi\)
−0.839525 + 0.543321i \(0.817167\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1548.2.f.b.773.3 ✓ 8
3.2 odd 2 inner 1548.2.f.b.773.6 yes 8
4.3 odd 2 6192.2.l.i.2321.6 8
12.11 even 2 6192.2.l.i.2321.3 8
43.42 odd 2 CM 1548.2.f.b.773.3 ✓ 8
129.128 even 2 inner 1548.2.f.b.773.6 yes 8
172.171 even 2 6192.2.l.i.2321.6 8
516.515 odd 2 6192.2.l.i.2321.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1548.2.f.b.773.3 ✓ 8 1.1 even 1 trivial
1548.2.f.b.773.3 ✓ 8 43.42 odd 2 CM
1548.2.f.b.773.6 yes 8 3.2 odd 2 inner
1548.2.f.b.773.6 yes 8 129.128 even 2 inner
6192.2.l.i.2321.3 8 12.11 even 2
6192.2.l.i.2321.3 8 516.515 odd 2
6192.2.l.i.2321.6 8 4.3 odd 2
6192.2.l.i.2321.6 8 172.171 even 2